Density of states for banded and sparse random matrices

The average density of states for a large class of N × N banded and sparse random matrices is shown to obey a semi-circle law. The banded matrices belonging to this class are restricted in several ways: 1) they are both real and symmetric, 2) matrix elements are independent random variables with zero average, 3) the variance of the matrix elements, Equation Found. decays monotonically away from the diagonal, 4) depends on i—j alone and the range over which it significantly varies, δy, satisfies 1« δy « N. On the other hand, the sparse matrices for which this results, are obtained by permuting the variance in each of the rows of the banded matrices.